Can you put 3-15 amp 2- 20 amp 2- 30 amp and 1-60 amp breakers on a 100 amp service?

Yes, the assumption is that when house circuitry is layed out not all appliances will be on at the same time. If for some strange reason they were all loaded up the main breaker would trip. Just keep in mind that the main purpose of all breakers is to protect the wire that is connected to it.

Secondally to remove any short circuits that might occur on these wires.

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